Another approach to notating JI

It occured to me that if we notate ennealimmal or hemiennealimmal, we effectively notate JI up to the 7 (or 11) limit. The only problem is, JI enthusthiasts would probably never believe in it or accept it.

The idea is to take an ordinary five-line staff, and have each line and space on it correspond to one step of 9-et. We would not use the space between staves as a space for a note, so that the bottom line of a treble staff would be exactly an octave above the bottom line of a bass staff. We would then use sharp to mean 21/20, and flat 20/21.
This suffices for the 7-limit; for the 11-limit, we would introduce a symbol meaning exactly half of a 9-et step; i.e., 1/18-th of an octave. We could of course use multiple sharp and flat symbols, and introduce a symbol which is a compliment to the sharp--that is, a step up and a flat down; or almost precisely 36/35, which could be useful though it isn't necessary.

Any thoughts? It seems like an awfully easy way to get the job done.

On Tue, 25 Jun 2002 10:43 +0100 (BST) gbreed@cix.compulink.co.uk (Graham
Breed) writes:

> Yes, but we can still notate the same limits uniquely using miracle,
> but
> much more efficiently. Why would ennealimmal be an advantage?

Because it is far more accurate, and hence can claim to be effecively JI.
Miracle doesn't really work to notate JI, but this is so close you
essentially can't tell the difference.

> What's
> hemiennealimmal?

Like ennalimmal, only dividing the octave in 18 parts, and then using
this to go to the 11-limit with extreme accuracy.
[[18, 30, 44, 52, 63], [0, -2, -3, -2, -1]] would be one way to do the
mapping.

On Tue, 25 Jun 2002 12:15:38 +0200 manuel.op.de.coul@eon-benelux.com
writes:
>
> Very nice, I will make a Scala notation for it, and call it EL72
> (EL=ennealimmal). Note names C D E F G H J A B C and ascii
> accidentals:

Looks good; one of the things I like about it is that you can notate with
ordinary musical notation programs such as Noteworthy.

> This can also be used for 27-tET. Maybe I'll make a EL99 too.
> Rather than introducing an extra symbol for 36/35, I thought
> that 7/ is fine too.

One might find extra symbols for commas such as 126/125=1728/1715, etc.
useful as well.

To clarify what I said about this being "much more accurate" than
miracle, we are talking about an order of magnitude difference, resulting
in something which basically is indistinguishable from JI.

On Tue, 25 Jun 2002 12:15:38 +0200 manuel.op.de.coul@eon-benelux.com
writes:
>
> Very nice, I will make a Scala notation for it, and call it EL72
> (EL=ennealimmal). Note names C D E F G H J A B C and ascii
> accidentals:
>
> 0: 1/1 C
> 1: 16.667 cents C/ DbL
> 2: 33.333 cents C7 Db\
> 3: 50.000 cents C7/ Db
> 4: 66.667 cents C#\ Db/
> 5: 83.333 cents C# DL\
> 6: 100.000 cents C#/ DL
> 7: 116.667 cents C#7 D\
> 8: 133.333 cents D

133.333 cents is right for the 1/9-octave steps, but these intermediate
steps ought to be concocted out of
21/20 = 84.467 cents, or something close to it, so this and not 83.333 is
what I was thinking of as C#; this 72-et
version may be good for naming things, but what I was thinking of is, as
I said, something an order of magnitude more
accurate--at least, if needed.

On Tue, 25 Jun 2002 11:45 +0100 (BST) gbreed@cix.compulink.co.uk (Graham
Breed) writes:
> In-Reply-To: <20020625.025934.-1949755.0.genewardsmith@juno.com>
> Gene W Smith wrote:

> It doesn't matter how accurate the temperament is, as long as the
> notation
> can uniquely identify each just interval.

Ennealimmal is accurate enough that it doesn't need to do this.

> I've added it to the catalog. It's accurate but complex. On a 9
> note
> staff, you need an accidental to get the 18 notes and 3 more to get
> all
> the intervals, giving 8 different combinations.

My system only needs a single accidental.

>Hemiennealimmal notation doesn't seem to
> offer any
> advantages unless you're using the temperament.

The point is, you do use temperament--you temper out 2401/2400 and
4375/4374, and nobody misses them.

why are all these posts that gene is replying to not showing up in
the archives?

On Tue, 25 Jun 2002 13:25 +0100 (BST) gbreed@cix.compulink.co.uk (Graham
Breed) writes:

> Gene:
> > Ennealimmal is accurate enough that it doesn't need to do this.
>
> If you make the colossal assumption that your performers can
> accurately
> reproduce such a temperament without using a just reference.

Why worry about a just preference? We won't know how this works as a
performance
system unless it is tried, but it does locate the JI intervals, and I
doubt very much adding goofy accidentals
for 2401/2400 and 4375/4374, making it a "true" JI system, would help
any.

> > My system only needs a single accidental.
>
> If you can get an 11-limit otonality with 9 nominals and 1
> accidental,
> that is good. I'll believe it when I see it.

18 nominals and 1 accidental for the 11-limit; 9 for the 7-limit.

> Then you're notating a temperament. We don't know what commas will
> be
> missed because we can't get a straight answer out of the JI
> proponents.

The answers we seem to be getting from those who are willing to answer
tell
us that the difference between ennealimmal and JI is not perceptible in
practice.

On Wed, 26 Jun 2002 10:54 +0100 (BST) gbreed@cix.compulink.co.uk (Graham
Breed) writes:
> In-Reply-To: <20020625.162929.-1949755.4.genewardsmith@juno.com>

> I said reference, not preference. If you want them to locate JI
> intervals, surely it's easier to teach them JI intervals than 9
> equally
> spaced notes where they have to add or subtract exactly 17.6 cents
> to get
> the desired intervals.

How do they locate these intervals? With a ratio? A collection of
accidentals? It seems
to me that you can hardly complain that my system is too complicated,
since any JI system will have the problem
of wishing to get things very exactly in tune, and this method at least
is not confusing the issue with commas which would not be relevant in
performance practice. We could take a JI score, at least up to the
11-limit, and notate it in a fairly straightforward way.

> I don't believe you can notate 7-limit intervals with one accidental
> the
> way I was counting them. Show an example of 4:5:6:7 if you think it
> can
> be done.

A B### E## G##

On Wed, 26 Jun 2002 12:58 +0100 (BST) gbreed@cix.compulink.co.uk (Graham
Breed) writes:
> In-Reply-To: <20020626.033725.-1949755.7.genewardsmith@juno.com>

> Yes, it has the same problems as any other JI system. It isn't "an
> awfully easy way to get the job done" as you originally said. If
> you're
> score uses Partch's 43 note scale (which is the best historical
> precedent)
> you only need to train the performers to produce those 43 notes by
> my
> calculations, if you're using a JI notation.

And then, that's all they can do. They can't even modulate worth a damn.

> A slightly less obvious way of getting this accuracy is to use a
> miracle
> generator of 116.59 cents and a schisma of around 2.2 cents. That
> can
> give you the 11-limit to around 0.3 cents.

If you really use miracle an schismic at the same time as temperaments,
you get the 41-et, so this won't be a logical system. If you want to add
symbols, it seems to me better to start out with something consistent.
Instead of a double sharp symbol, we could add two sharps up and a step
down, for a symbol covering 49/48 and 50/49. Then two steps up and three
flats down give us a symbol covering 245/243, 126/125, 4000/3969 and
1728/1715 which we can use in place of a special three sharps/three flats
symbol. Putting these together would give us a five sharps/five flats
symbol covering 875/864, 81/80, 3125/3087 and 2430/2401. Putting this
together with the three sharps symbol gives us eight sharps up and five
steps down, covering 1029/1024, 225/224, 19683/19600 and
16875/16807--miracle commas. Setting this comma to unison puts you into
72-et, which you would now be notating.

On Wed, 26 Jun 2002 12:58 +0100 (BST) gbreed@cix.compulink.co.uk (Graham
Breed) writes:

> A slightly less obvious way of getting this accuracy is to use a
> miracle
> generator of 116.59 cents and a schisma of around 2.2 cents. That
> can
> give you the 11-limit to around 0.3 cents.

One way to do this would be via the 494-et, where 48/494 was one symbol,
and 1/494 was another.
The 494-et also convers enneadecal (with map [[19,30,44,54],[0,1,1,3]])
and hemienneadecal
(with map [[38,60,88,106,131],[0,1,1,3,2]]) where we can play the same
kind of game, only with 19 notes instead of 9.
It's a microtemperament also, and gets things quite accurately, but I
don't see any advantage to it.

In-Reply-To: <20020626.053859.-1949755.8.genewardsmith@juno.com>
Me:
> > Yes, it has the same problems as any other JI system. It isn't "an
> > awfully easy way to get the job done" as you originally said. If
> > you're
> > score uses Partch's 43 note scale (which is the best historical
> > precedent)
> > you only need to train the performers to produce those 43 notes by
> > my
> > calculations, if you're using a JI notation.

Gene:
> And then, that's all they can do. They can't even modulate worth a damn.

But that's what they *want" to do. If they wanted to do something else
they could have learned that instead. The system already contains room
for modulation. You can do a fair bit more without needing 234 notes. If
you are doing that much modulation, it won't be a typical JI composition,
judging by what people are saying in another place.

Me:
> > A slightly less obvious way of getting this accuracy is to use a
> > miracle
> > generator of 116.59 cents and a schisma of around 2.2 cents. That
> > can
> > give you the 11-limit to around 0.3 cents.

Gene:
> If you really use miracle an schismic at the same time as temperaments,
> you get the 41-et, so this won't be a logical system.

That isn't at all what I suggested, so it's hardly a relevant objection.

> If you want to add
> symbols, it seems to me better to start out with something consistent.

The system's perfectly consistent, before and after adding the schisma.
It starts as

mapping by period:
[1, 1, 3, 3, 2]

mapping by generator:
[0, 6, -7, -2, 15]

and I've added

mapping by schisma:
[0, 1, 1, 1, 1]

to get a planar temperament.

> Instead of a double sharp symbol, we could add two sharps up and a step
> down, for a symbol covering 49/48 and 50/49. Then two steps up and three
> flats down give us a symbol covering 245/243, 126/125, 4000/3969 and
> 1728/1715 which we can use in place of a special three sharps/three
> flats
> symbol. Putting these together would give us a five sharps/five flats
> symbol covering 875/864, 81/80, 3125/3087 and 2430/2401. Putting this
> together with the three sharps symbol gives us eight sharps up and five
> steps down, covering 1029/1024, 225/224, 19683/19600 and
> 16875/16807--miracle commas. Setting this comma to unison puts you into
> 72-et, which you would now be notating.

Where's this starting from? 49:48 and 50:49 are quommas. So the "double
sharp symbol" is two steps of 72-equal? Double sharps would usually be 12
steps. With my modified system, I think 50:49 is the "true quomma" and
49:48 is a schisma wider.

Your second symbol is for 3 steps of 72-equal? I think your commas map as
follows to my schismas:

5 schismas =~ 19683:19600
4 schismas =~ 1029:1024
3 schismas =~ 225:224
2 schismas =~ 16875:16807

so they're all distinguished, but not in the right order (19683:19600 is
the second smallest). The single schisma miracle comma 2401:2400 is
around the same size.

Ah, "sharps" and "steps" seem to come from your hemiennialimmal notation.
So a 72 note hemiennialimmal tuning with a 7.82 cent "comma" shift? It'd
certainly be worth a try if we had a group of eager volunteers to try this
stuff out on.

Graham

In-Reply-To: <20020626.070552.-1949755.9.genewardsmith@juno.com>
Gene W Smith wrote:

> One way to do this would be via the 494-et, where 48/494 was one symbol,
> and 1/494 was another.
> The 494-et also convers enneadecal (with map [[19,30,44,54],[0,1,1,3]])
> and hemienneadecal
> (with map [[38,60,88,106,131],[0,1,1,3,2]]) where we can play the same
> kind of game, only with 19 notes instead of 9.
> It's a microtemperament also, and gets things quite accurately, but I
> don't see any advantage to it.

494-equal's nice to know about, well done. I don't see the point about
the enneadecals. Do you want them catalogued?

Graham

On Wed, 26 Jun 2002 15:24 +0100 (BST) graham@microtonal.co.uk writes:

> mapping by period:
> [1, 1, 3, 3, 2]
>
> mapping by generator:
> [0, 6, -7, -2, 15]
>
> and I've added
>
> mapping by schisma:
> [0, 1, 1, 1, 1]
>
> to get a planar temperament.

This is indeed a microtemperament, closely allied to the 494-et. I was
suggesting 48/494 and
1/494, and the rms generators for this temperament are 48.000/494 and
.971/494, so it seems we may as well
simply use 494 and be done with it for this one.

On Wed, 26 Jun 2002 15:24 +0100 (BST) graham@microtonal.co.uk writes:

If
> you are doing that much modulation, it won't be a typical JI
> composition,
> judging by what people are saying in another place.

I modulate quite a bit in electronic JI. People don't do more of it
because they can't, I suspect.

> Where's this starting from? 49:48 and 50:49 are quommas. So the
> "double
> sharp symbol" is two steps of 72-equal?

It comes from 2401/2400 being a comma. The sharp is 21/20, and a step is
27/25, so a
double sharp symbol represents (21/20)^2 / (27/25) = 49/48. It is two
72-et steps, five 171-et steps,
and 18 612-et steps.

> Your second symbol is for 3 steps of 72-equal?

No, for one step. A triple flat symbol is (27/25)^2 / (21/20)^3 =
1728/1715, which is also 126/125, etc.
The symbols are additive, and would come in a Fibbonaci set: {1, 2, 3, 5,
8}.

I think your commas
> map as
> follows to my schismas:

> 5 schismas =~ 19683:19600
> 4 schismas =~ 1029:1024
> 3 schismas =~ 225:224
> 2 schismas =~ 16875:16807

These are all the same (being eight sharps) in my system, so the two
systems seem to be very different.

> Ah, "sharps" and "steps" seem to come from your hemiennialimmal
> notation.
> So a 72 note hemiennialimmal tuning with a 7.82 cent "comma" shift?

That's one way to look at it.

> It'd
> certainly be worth a try if we had a group of eager volunteers to
> try this
> stuff out on.

Always the rub. I suppose I could notate some Partch, claim it was JI,
and see the fur fly over on the main group.

--- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:
> Always the rub. I suppose I could notate some Partch, claim it was
> JI, and see the fur fly over on the main group.

Like we don't read this list? :)

Cheers,
Jon

--- In tuning-math@y..., "jonszanto" <jonszanto@y...> wrote:
> --- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:
> > Always the rub. I suppose I could notate some Partch, claim it was
> > JI, and see the fur fly over on the main group.

> Like we don't read this list? :)

Only on your best behavior. :)